A congruence of \(\mathbf {A}\)  is the kernel of a homomorphism
with domain \(\mathbf {A}\) ; equivalently, a congruence of \(\mathbf {A}\) 
is an equivalence relation on \(A\) 
that is invariant under the
operations of \(\mathbf {A}\) .
If \(\vartheta \)  is a congruence of \(\mathbf {A}\) , then there is a unique
algebra, denoted \(\mathbf {A}/\vartheta \) , whose universe is
the set
\(A/\vartheta \)  of equivalence classes of \(\vartheta \) , and whose operations
are defined so that the natural map \(\nu \colon A\rightarrow A/\vartheta \) ,
\(a\mapsto a/\vartheta \)  is a homomorphism \(\mathbf {A}\rightarrow \mathbf {A}/\vartheta \) .
Here \(a/\vartheta \)  denotes
the equivalence class of \(\vartheta \)  containing \(a\) .
The algebra \(\mathbf {A}/\vartheta \)  is called a
quotient algebra of \(\mathbf {A}\) .
By the first
isomorphism theorem ([1]}),
if \(\varphi \colon \mathbf {A}\rightarrow \mathbf {B}\) 
is an onto homomorphism with kernel \(\vartheta \) , then \(\mathbf {B}\)  is isomorphic
to \(\mathbf {A}/\vartheta \) .
